Question

Number of the ordered pair (s) satisfying the system of equations
$2\log (x^2+y^2)-\log 5=\log \left\{2\right(x^2+y^2)+75\}\text{and log}\left(\frac{x}{3}\right)+\log \left(5y\right)=1+\log 2$ is

• A. $1$
• B. $2$
• C. $3$
• D. $4$

Answer 1

The correct option is B $2$

$2\log (x^2+y^2)-\log 5=\log \left\{2\right(x^2+y^2)+75\}$

$⟹\log \frac{(x^2+y^2{)}^2}{5}=\log \left(2\right(x^2+y^2)+75)$

$⟹(x^2+y^2{)}^2-10(x^2+y^2)-375=0$

$x^2+y^2=25,-15$ but sum of $2$ square can't be $-15$.

Hence

$x^2+y^2=25=5^2..............\left(1\right)$ [equation of a circle]

Now,

$\log \frac{x}{3}+\log 5y=1+\log 2$

$⟹\frac{5xy}{3}=20$.........[since, $1=\log 10$]

$⟹xy=12$.............($2$)

Solving $1$ and $2$ we get:

$(x,y)=(3,4)or(4,3)$

Hence the number of ordered pair is $2$